Adjoint of Mesh Deformation

The ultimate sensitivities of a given objective, or cost function, are those with respect to the design parameters, that is $\frac{d L}{d D}$ . These sensitivities are typically used to solve a shape optimization problem.

Design parameters, $D$ , define the mesh coordinates of a shape, $X (D)$ . A mesh deformation strategy aims at deforming the initial baseline mesh of a shape, $X^{0}$ , based on the design parameters to evolve new shapes, $X (X^{0}, D)$ . The mesh deformation algorithm that is used for the adjoint method in Simcenter STAR-CCM+ is the radial basis function (RBF) morpher algorithm. The adjoint derivative of the morpher algorithm evaluates:

Figure 1. EQUATION_DISPLAY

\frac{d L^{T}}{d D} = \frac{d X^{T}}{d D} \frac{d L^{T}}{d X}

(5087)

where $\frac{d L^{T}}{d X}$ is the provided final mesh sensitivity.

In a shape optimization procedure, the computed design parameter sensitivity as given by Eqn. (5087) is the basis for the calculation of the new design parameter values. The morpher deforms the mesh coordinates of the shape according to these new values.

Primal

Using radial basis functions, the interpolated position $x$ about the reference position $x^{0}$ is given by

Figure 2. EQUATION_DISPLAY

x (x^{0}) = α + \sum_{j = 1}^{N} β_{j} ϕ_{j} (r_{j} (x^{0}))

(5088)

where

Figure 3. EQUATION_DISPLAY

r_{j} (x^{0}) = ‖ x^{0} - x_{j}^{0} ‖

(5089)

There are two types of radial basis function: those with compact support and those with global support. Here, only one global support function is considered, the multiquadratic biharmonics function $ϕ (r)$ given as:

Figure 4. EQUATION_DISPLAY

ϕ_{j} (r_{j}) = \sqrt{r_{j}^{2} + c_{j}^{2}}

(5090)

and the constant $c_{j} = 0, \forall j$ . For a discrete set of $N$ reference positions in the domain, evaluation of all $M$ interpolation positions in that domain can be evaluated by casting Eqn. (5088) as:

Figure 5. EQUATION_DISPLAY

[\begin{matrix} x_{1} \\ \begin{matrix} \begin{matrix} ⋮ \end{matrix} \\ {\begin{matrix} x \end{matrix}}_{M} \end{matrix} \end{matrix}] = [\begin{matrix} 1_{1} \\ \begin{matrix} \begin{matrix} ⋮ \end{matrix} \\ 1_{M} \end{matrix} \end{matrix}] α + [\begin{matrix} ϕ_{1, 1} & \dots & ϕ_{1, N} \\ ⋮ & ⋱ & ⋮ \\ ϕ_{M, 1} & \dots & ϕ_{M, N} \end{matrix}] [\begin{matrix} β_{1} \\ \begin{matrix} \begin{matrix} ⋮ \end{matrix} \\ β_{N} \end{matrix} \end{matrix}]

(5091)

where the components of the $N \times M$ matrix are computed using the radial basis function

Figure 6. EQUATION_DISPLAY

ϕ_{i, j} = ϕ_{j} (r_{j} (x_{i}^{0}))

(5092)

The continuous interpolation function Eqn. (5088) or its discrete form of Eqn. (5091) require the coefficients $α$ and $β_{1}, ..., β_{N}$ . These coefficients are determined by $N$ known interpolation positions, or design parameters, $D = d_{1}, ..., d_{N}$ , and the additional constraint that

Figure 7. EQUATION_DISPLAY

\sum_{j = 1}^{N} β_{j} = 0

(5093)

From Eqn. (5088) and Eqn. (5093), a system of equations is constructed to solve the unknown coefficients:

Figure 8. EQUATION_DISPLAY

[\begin{matrix} ϕ_{1, 1} & \dots & ϕ_{1, N} & 1_{1} \\ ⋮ & ⋱ & ⋮ & ⋮ \\ ϕ_{N, 1} & \dots & ϕ_{N, N} & 1_{N} \\ 1_{1} & \dots & 1_{N} & 0 \end{matrix}] [\begin{matrix} β_{1} \\ ⋮ \\ {\begin{matrix} β \end{matrix}}_{N} \\ α \end{matrix}] = [\begin{matrix} d_{1} \\ ⋮ \\ d_{N} \\ 0 \end{matrix}]

(5094)

Adjoint

From a functional point of view, the independent variables of the morpher are $D = d_{1}, ..., d_{N}$ , and its dependent variable is $x$ in the continuous form and $X = x_{1}, ..., x_{M}$ in the discrete form. The positions $X^{0}$ are constants of the system. Since only the discrete form of the morpher is of interest, the required Jacobian is then:

Figure 9. EQUATION_DISPLAY

J_{X} = \frac{d X}{d D}

(5095)

A tangent, $j$ , of the Jacobian Eqn. (5095) is derived by applying the chain rule to Eqn. (5091):

Figure 10. EQUATION_DISPLAY

\frac{\partial X}{\partial d_{j}} = [\begin{matrix} \frac{\partial x_{1}}{\partial d_{j}} \\ \begin{matrix} \begin{matrix} ⋮ \end{matrix} \\ \frac{\partial x_{M}}{\partial d_{j}} \end{matrix} \end{matrix}] = [\begin{matrix} 1_{1} \\ \begin{matrix} \begin{matrix} ⋮ \end{matrix} \\ 1_{M} \end{matrix} \end{matrix}] \frac{\partial α}{\partial d_{j}} + [\begin{matrix} ϕ_{1, 1} & \dots & ϕ_{1, N} \\ ⋮ & ⋱ & ⋮ \\ ϕ_{M, 1} & \dots & ϕ_{M, N} \end{matrix}] [\begin{matrix} \frac{\partial β_{1}}{\partial d_{j}} \\ \begin{matrix} \begin{matrix} ⋮ \end{matrix} \\ \frac{\partial β_{N}}{\partial d_{j}} \end{matrix} \end{matrix}]

(5096)

The derivative of the coefficients $α$ and $β_{1}, ..., β_{N}$ with respect to $d_{j}$ are determined by solving the following system:

Figure 11. EQUATION_DISPLAY

[\begin{matrix} ϕ_{1, 1} & \dots & ϕ_{1, N} & 1_{1} \\ ⋮ & ⋱ & ⋮ & ⋮ \\ ϕ_{N, 1} & \dots & ϕ_{N, N} & 1_{N} \\ 1_{1} & \dots & 1_{N} & 0 \end{matrix}] [\begin{matrix} \frac{\partial β_{1}}{\partial d_{j}} \\ ⋮ \\ \frac{\partial β_{N}}{\partial \begin{matrix} d_{j} \end{matrix}} \\ \frac{\partial α}{\partial d_{j}} \end{matrix}] = [\begin{matrix} \frac{\partial d_{1}}{\partial d_{j}} \\ ⋮ \\ \frac{\partial d_{N}}{\partial \begin{matrix} d_{j} \end{matrix}} \\ 0 \end{matrix}]

(5097)

Therefore, the requirement for solving the tangent derivative is the capacity to evaluate Eqn. (5091) and Eqn. (5094) for specific inputs.

The aim of the adjoint evaluation of the morpher is to compute Eqn. (5087). The second term on the right hand side of this equation is already provided (see Sensitivity of Cost Function w.R.t. Flow and Energy Solution), so the adjoint of the morpher must compute the product of the two terms, that is, ${\frac{d L}{d D}}^{T}$ .

This product is derived by applying the chain rule to Eqn. (5091) and transposing the result:

Figure 12. EQUATION_DISPLAY

\begin{matrix} \frac{\partial L}{\partial \begin{matrix} α \end{matrix}} = {[\begin{matrix} 1_{1} \\ ⋮ \\ 1_{M} \end{matrix}]}^{T} [\begin{matrix} \frac{\partial L}{\partial x_{1}} \\ ⋮ \\ \frac{\partial L}{\partial x_{M}} \end{matrix}] \\ [\begin{matrix} \frac{\partial L}{\partial β_{1}} \\ \begin{matrix} \begin{matrix} ⋮ \end{matrix} \\ \frac{\partial L}{\partial β_{N}} \end{matrix} \end{matrix}] = {[\begin{matrix} ϕ_{1, 1} & \dots & ϕ_{1, N} \\ ⋮ & ⋱ & ⋮ \\ ϕ_{M, 1} & \dots & ϕ_{M, N} \end{matrix}]}^{T} [\begin{matrix} \frac{\partial L}{\partial x_{1}} \\ \begin{matrix} \begin{matrix} ⋮ \end{matrix} \\ \frac{\partial L}{\partial x_{M}} \end{matrix} \end{matrix}] \end{matrix}

(5098)

On evaluating Eqn. (5098), the adjoint is obtained by solving the following system:

Figure 13. EQUATION_DISPLAY

{[\begin{matrix} ϕ_{1, 1} & \dots & ϕ_{1, N} & 1_{1} \\ ⋮ & ⋱ & ⋮ & ⋮ \\ ϕ_{N, 1} & \dots & ϕ_{N, N} & 1_{N} \\ 1_{1} & \dots & 1_{N} & 0 \end{matrix}]}^{T} [\begin{matrix} \frac{\partial L}{\partial d_{1}} \\ ⋮ \\ \frac{\partial L}{\partial \begin{matrix} d_{N} \end{matrix}} \\ t \end{matrix}] = [\begin{matrix} \frac{\partial L}{\partial β_{1}} \\ ⋮ \\ \frac{\partial L}{\partial \begin{matrix} β_{N} \end{matrix}} \\ \frac{\partial L}{\partial α} \end{matrix}]

(5099)

where the result $t$ is discarded.